Spring 2016 Mathematics W4053 section 001 Columbia University, Department of Mathematics Day/Time: Tuesdays&Thursdays 5:40pm-6:55pm Location: 520 Mathematics Building Instructor: Anton Zeitlin |

Office hours: Monday, 2 pm-3 pm and by appointment |

Grade distribution: 30% homeworks, 30% Midterm, 40 % Final |

Midterm: Tuesday, March 8 |

Final exam: Take-home |

TA: Kat Christianson |

TA office hours: 4-5pm on Tuesdays and 12-1pm on Thursdays |

**Approximate Syllabus.**

**Part I. Basic homotopy theory**

- Homotopy and homotopy equivalence.
- Fundamental group and higher homotopy groups.
- CW complexes.
- Calculation of the fundamental group, van Kampen theorem.
- Covering spaces and their classification.
- Relative homotopy groups, long exact sequences of pairs.
- Fibrations, long exact sequence of a fibration.
- Manifolds, degree of a map.

**Part II. Homology**

- Chain complexes, elements of homological algebra.
- Homology of CW complexes.
- Homotopy and homology, Hurewicz theorem.
- Homology and Cohomology.
- Kunneth formula.
- Applications.
- Manifolds, Poincare duality.

**Lecture-to-lecture syllabus.** Will be updated in timely manner (with notes).

- Lecture I. Basic topological notions. Homotopy and and Homotopy equivalence (pdf).
- Lecture II. Fundamental group and higher fundamental groups. Basics (pdf).
- Lecture III. Van Kampen theorem (pdf).
- Lecture IV. CW complexes (pdf).
- Lecture V. Covering spaces (pdf).
- Lecture VI. Covering spaces (continued) (pdf).
- Lecture VII. Relative homotopy groups (pdf).
- Lecture VIII. Fibrations, fiber bundles (pdf).
- Lecture IX. Singular homology (pdf).
- Lecture X. Singular homology (continued) (pdf).
- Lecture XI. Eilenberg-Steenrod axioms (pdf).
- Lecture XII. Cellular homology (pdf).
- Lecture XIII. Degree of a map and cellular homology (pdf).
- Lecture XIV. Cohomology (pdf).
- Lecture XV. Projective modules and ext. functor (pdf).
- Lectures XVI-XVII. Cohomology ring, continued Poincare duality (pdf).
- Lecture XVIII. Separation theorems (pdf).
- Lecture XIX. Extra techniques for homotopy groups (pdf).
- Lecture XX. Hurewicz map, further technique (pdf).
- Lecture XXI. Smooth manifolds (some notions) (pdf).
- Lecture XXII. Elements of Morse theory (pdf).
- Lecture XXIII. Morse theory (continued) (pdf).

**Homeworks:**

Homework 2. pdf due February 18

Homework 3. pdf due March 3

Homework 4. pdf due March 31

Homework 5. pdf due April 14

**Recommended books:**

- V. A. Vassiliev
"Introduction to Topology"

http://www.amazon.com/Introduction-Topology-Student-Mathematical-Library/dp/0821821628 - A. Hatcher, "Algebraic topology", (free online book, standard textbook)

http://www.math.cornell.edu/~hatcher/AT/ATpage.html - A.Fomenko, D.Fuchs, and V.Gutenmacher, "Homotopic Topology", (old russian textbook, also very useful)

http://www.math.columbia.edu/~khovanov/algtop2013/Fuchs1.pdf